Originally written 14 May 1993 for the Extropians
mailing list, and re-published with 2001 at the urging of a list
member, with light editing by Melinda
Green. Reformatted and with minor fixes December 2005.

Abstract:

This essay discusses the best current understanding of
the relationship between mathematical and empirical
knowledge. It focuses on two questions:

Does mathematics have some sort of deep metaphysical connection
with reality, and

if not, why is it that mathematical abstractions seem so often
to be so powerfully predictive in the real world?

Mathematics is the model of a-priori knowledge in the
Aristotelian tradition of rationalism. Among the Greeks, geometry
was regarded as the highest form of knowledge, a potent key to the
metaphysical mysteries of the Universe. This is a rather
mystical belief, and the connection to mysticism and religion was
made explicit in cults like the Pythagorean. No culture since
has semi-deified a man for discovering a geometrical theorem!

The Greek awe of mathematical knowledge is still with us; it's
behind the traditional metaphor of mathematics as "Queen of the
Sciences". It's been reinforced by the spectacular successes
of mathematical models in science, successes the Greeks (lacking
even simple algebra) could never have foreseen. Since Isaac
Newton's discovery of calculus and the inverse-square law of
gravity in the late 1600s, phenomenal science and higher
mathematics have been closely symbiotic -- so much so, that the
existence of a predictive mathematical formalism has become the
hallmark of a "hard science".

For two centuries after Newton, phenomenal science aspired to
the kind of rigor and purity that seemed to be embodied in
mathematics. The metaphysical situation seemed simple;
mathematics embodied perfect a-priori knowledge, those sciences
able to most mathematicize themselves were the most successful at
phenomenal prediction; perfect knowledge would therefore consist of
a mathematical formalism, arrived at by science and embracing all
of reality, that would ground a-posteriori empirical understanding
in a-priori rational logic. It was in this spirit that
Condorcet dared to imagine describing the entire universe as a
mutually-solving set of partial differential equations.

The first cracks in this inspiring picture appeared in the
latter half of the 19th century when Riemann and Lobachevsky
independently proved that Euclid's Axiom of Parallels could be
replaced by alternatives which yielded consistent geometries.
Riemann's geometry was modeled on a sphere, Lobachevsky's on a
hyperboloid of rotation.

The impact of this discovery has been obscured by later and
greater upheavals, but at the time it broke on the intellectual
world like a thunderbolt. For the existence of mutually
inconsistent axiom systems for geometry, any of which could be
modeled in the phenomenal universe, called the whole
relationship between mathematics and physical theory into
question.

When there was only Euclid, there was only one possible
geometry. One could believe that the Euclidean axioms
constituted a kind of perfect a-priori knowledge about geometry in
the phenomenal world. But suddenly we had three geometries,
an embarrassment of metaphysical riches.

For how were we to choose between the axioms of plane,
spherical, and hyperbolic geometry as a description of "real"
geometry? Because all three are consistent, we couldn't
choose on any a-priori basis -- the choice had to become empirical,
based on their predictive power for a given situation.

Of course, physical theorists had long been accustomed to
choosing formalisms to fit a scientific problem. But it had
been widely, if unconsciously, assumed that the need to do so ad
hoc was a function of human ignorance; that, given good enough
mathematics and logic, we could deduce the correct choice
from first principles, producing a-priori descriptions of reality
to be confirmed, as an afterthought, by empirical check.

But now, the Euclidean geometry that had been considered the
model for axiomatic perfection in mathematics for over two thousand
years, had been dethroned. If one could not know something as
fundamental as the geometry of space a-priori, what hope was there
for a purely "rational" theory encompassing all of nature?
Psychologically, Riemann/Lobachevsky struck at the very heart of
the enterprise of mathematics as it was then conceived.

Furthermore, Riemann/Lobachevsky called the nature of
mathematical intuition into question. It had been easy to
believe implicitly that mathematical intuition was a form of
perception -- a glimpse of the Platonic noumena behind
reality. But with two other geometries jostling Euclid,
nobody knew for sure what the noumena looked like any more!

Mathematicians responded to this dual problem with an increase
in rigor, by trying to apply the axiomatic method throughout
mathematics. It was gradually realized that the belief in
mathematical intuition as a kind of perception of a noumenal world
had encouraged sloppiness; proofs in the pre-axiomatic period often
relied on shared intuitions about mathematical "reality" that could
no longer be considered automatically valid.

The new thinking in mathematics led to a series of spectacular
successes; among these were Cantorian set theory, Frege's
axiomatization of number, and eventually Russell & Whitehead's
monumental synthesis in Principia Mathematica.

However, it also had a price. The axiomatic method made
the connection between mathematics and phenomenal reality narrower
and narrower. At the same time, discoveries like the
Banach-Tarski Paradox suggested that mathematical axioms that
seemed to be consistent with phenomenal experience could lead to
dizzying contradictions with that experience.

The majority of mathematicians quickly became "Formalists",
holding that pure mathematics could not be philosophically
considered more than a sort of elaborate game played with marks on
paper (this is the theory behind Robert Heinlein's pithy
characterization of mathematics as "a zero-content system"). The
old-fashioned "Platonist" belief in the noumenal reality of
mathematical objects seemed headed for the dustbin, despite the
fact that mathematicians continued to feel like Platonists
during the process of mathematical discovery.

Philosophically, then, the axiomatic method lead most
mathematicians to abandon previous beliefs in the metaphysical
specialness of mathematics. It also created today's split
between pure and applied mathematics.

Most of the great mathematicians of the early modern period --
Newton, Liebniz, Fourier, Gauss, and others -- were also phenomenal
scientists (i.e. "natural philosophers"). The axiomatic
method incubated the modern idea of the pure mathematician as
super-esthete, unconcerned with the merely physical.
Ironically, Formalism gave pure mathematicians a bad case of
Platonic attitude. Applied mathematicians stopped being
invited to tea and learned to hang out with physicists.

This brings us to the early 20th century. For the beleaguered
minority of Platonists, worse was yet to come.

Cantor, Frege, Russell and Whitehead showed that all of pure
mathematics could be built on the single axiomatic foundation of
set theory. This suited the Formalists just fine; mathematics
coalesced, at least in principle, from a bunch of little
disconnected games to one big game. It also made the
Platonist minority happy; if there turned out to be one big,
over-arching consistent structure behind all of mathematics, the
metaphysical specialness of mathematics might yet be rescued.

Unfortunately, it turns out that there is more than one way to
axiomatize set theory. In particular, there are at least four
major different combinations of assumptions about infinite sets
that lead to mutually exclusive set theories (the Axiom of Choice
or its negation; the Continuum Hypothesis or its negation).

It was Riemmann/Lobachevsky all over again, but on a much more
fundamental level. Riemannian and Lobachevskian geometry
could be modeled finitely, in the world; you could decide at least
empirically which one fit. Normally, you could regard all
three as special cases of the geometry of geodesics on manifolds,
thereby fitting them into the superstructure erected on set
theory.

But the independent axioms in set theory don't seem to lead to
any results that can be modeled in the observable finite
world. And there's no way to assert both the Continuum
Hypothesis and its negation in one set theory. How's a poor
Platonist to choose which system describes "real"
mathematics? The victory of the Formalist position seemed
complete.

In a negative way, though, a Platonist had the last laugh.
Kurt Godel threw a spanner in the Formalist program of
axiomatization when he showed that any axiom system powerful enough
to include the integers would have to be either inconsistent
(yielding contradictions) or incomplete (too weak to decide the
truth or falsehood of some assertions in the system).

And that is more or less where things stand today.
Mathematicians know that any attempt to put forward mathematics as
a-priori knowledge about the universe must fall afoul of numerous
paradoxes and impossible choices about what axiom system describes
"real" mathematics. They've been reduced to hoping that the
standard axiomatizations are not inconsistent but incomplete, and
wondering uneasily what contradictions or unprovable theorems are
waiting discovery out there, lurking like landmines in the
noosphere.

Meanwhile, on the empirical front, mathematics continued to be a
spectacular success as a theory-building tool. The great
successes of 20th-century physics (general relativity and quantum
mechanics) wandered so far from the realm of physical intuition
that they could be understood only by meditating deeply on their
mathematical formalisms, and following through to their logical
conclusions, even when those conclusions seem wildly bizarre.

What irony. Even as mathematical 'perception' came to seem
less and less reliable in pure mathematics, it became more and more
indispensable in phenomenal science!

Against this background, Einstein's famous quote wondering at
the applicability of mathematics to phenomenal science poses an
even thornier problem than at first appears.

The relationship between mathematical models and phenomenal
prediction is complicated, not just in practice but in
principle. Much more complicated because, as we now know,
there are mutually exclusive ways to axiomatize mathematics!
It can be diagrammed as follows (thanks to Jesse Perry for supplying the
original of this chart):

The key transactions for our purposes are C
and D -- the translations between a predictive
model and a mathematical formalism. What mystified Einstein
is how often D leads to new insights.

We begin to get some handle on the problem if we phrase it more
precisely; that is, "Why does a good choice of C
so often yield new knowledge via D?"

The simplest answer is to invert the question and treat it as a
definition. A "good choice of C" is one
which leads to new predictions. The choice of C is not one that can be made a-priori; one has to
choose, empirically, a mapping between real and mathematical
objects, then evaluate that mapping by seeing if it predicts
well.

For example, the positive integers are a good formalism for
counting marbles. We can confidently predict that if we put two
marbles in a jar, and then put three marbles in a jar, and then
empirically associate the set of two marbles with the mathematical
entity 2, and likewise associate the set of three
marbles with the mathematical entity 3, and then
assume that physical aggregation is modeled by +,
then the number of marbles in the jar will correspond to the
mathematical entity 5.

The above may seem to be a remarkable amount of pedantry to load
on an obvious association, one we normally make without having to
think about it. But remember that small children have to
learn to count...and consider how the above would fail if
we were putting into the jar, not marbles, but lumps of mud or
volumes of gas!

One can argue that it only makes sense to marvel at the utility
of mathematics if one assumes that C for any
phenomenal system is an a-priori given. But we've seen that
it is not. A physicist who marvels at the applicability of
mathematics has forgotten or ignored the complexity of C; he is really being puzzled at the human
ability to choose appropriate mathematical models
empirically.

By reformulating the question this way, we've slain half the
dragon. Human beings are clever, persistent apes who like to
play with ideas. If a mathematical formalism can be found to fit a
phenomenal system, some human will eventually find it. And
the discovery will come to look "inevitable" because those who
tried and failed will generally be forgotten.

But there is a deeper question behind this: why do good choices
of mathematical model exist at all? That is, why is
there any mathematical formalism for, say, quantum
mechanics which is so productive that it actually predicts the
discovery of observable new particles?

The way to "answer" this question is by observing that it, too,
properly serves as a kind of definition. There are many
phenomenal systems for which no such exact predictive formalism has
been found, nor for which one seems likely. Poets like to
mumble about the human heart, but more mundane examples are
available. The weather, or the behavior of any economy larger
than village size, for example -- systems so chaotically
interdependent that exact prediction is effectively impossible (not
just in fact but in principle).

There are many things for which mathematical modeling leads at
best to fuzzy, contingent, statistical results and never
successfully predicts 'new entities' at all. In fact, such
systems are the rule, not the exception. So the proper answer
to the question "Why is mathematics is so marvelously applicable to
my science?" is simply "Because that's the kind of science you've
chosen to study!"